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6th Grade Math Worksheet - A Parent's Guide

Many educators, politicians, and parents believe the instruction of mathematics in the United States is in crisis mode, and
has been for some time. Indeed, recent test results show that American 15-year-olds were
outperformed by 29 other countries on math testing scores.
1 To help counter this crisis, educational, civic, and business leaders worked together
to develop the Common Core State Standards (CCSS).

The goal of Common Core is to establish consistent, nationwide guidelines of what children should be learning each school
year, from kindergarten all the way through high school, in English and math. Though CCSS
sets forth these criteria, states and school districts are tasked with developing curricula
to meet the standards.

The 2014-15 school year will be important for Common Core as the standards are fully implemented in many remaining states
of the 43 (and the District of Columbia) that have embraced their adoption. CCSS has its
advocates as well as its critics, and the debate on its merits has become more pronounced
in recent months. Irrespective of the political differences with Common Core, its concepts
are critical for students because the standards help with understanding the foundational
principles of how math works. This guide steers clear of most of the controversy surrounding
CCSS and primarily focuses upon the math your sixthgrader will encounter.

Common Core Standards

A stated objective of Common Core is to standardize academic guidelines nationwide. In other words, what sixth-graders are
learning in math in one state should be the same as what students of the same age are learning
in another state. The curricula may vary between these two states, but the general concepts behind
them are similar. This approach is intended to replace wildly differing guidelines among different
states, thus eliminating (in theory) inconsistent test scores and other metrics that gauge student
success.

An increased focus on math would seem to include a wider variety of topics and concepts being taught at every grade level,
including sixth grade. However, CCSS actually calls for fewer topics at each grade level. The
Common Core approach (which is clearly influenced by so-called “Singapore Math”—an educational
initiative that promotes mastery instead of memorization) goes against many state standards,
which mandate a “mile-wide, inch-deep” curriculum in which children are being taught so much
in a relatively short span of time that they aren’t effectively becoming proficient in the concepts
they truly need to succeed at the next level. Hence, CCSS works to establish an incredibly thorough
foundation not only for the math concepts in future grades, but also toward practical application
for a lifetime.

For sixth grade, Common Core’s focus includes ratios, division of fractions, negative numbers, and variables. Ultimately,
this focus will enable children to develop rigor in reallife situations by developing a base
of conceptual understanding and procedural fluency.

Critical Areas of Focus

Fasten your seatbelts: Sixth-graders are in for a wild ride this year in math. Many new concepts, such as negative numbers
and variables, are introduced, but students’ previous learning will set them up nicely for learning
these topics. Furthermore, much of what debuts in sixth grade provides some foundation for the
algebra in the near future. Here are the four critical areas that Common Core brings to sixth-grade
math:

Ratio

Students not only apply their proficiency of multiplication and division that they learned in earlier grades, they also use
fractions extensively to solve problems about ratios and rates (e.g, if a recipe uses 4 cups
of flour to make 20 cookies, how many cups are needed to make 5 cookies?).

Division of Fractions, Negative Numbers

Negative numbers are introduced, with an emphasis on negative rational numbers, negative integers, and absolute value. In
the last of the four basic operations they will apply to fractional equations, students learn
to divide fractions by fractions. The graph system, on which students previously were working
with only one quadrant, is expanded to include all four quadrants on the coordinate plane.

Expressions and Equations

In what may be the strongest preview of future algebra, students will learn to solve one-step equations using variables (e.g,
x+10=17, solve for x). Sixth-graders will also rewrite equations in equivalent forms and understand
that a solution is the values of the variables that make an equation true.

Statistical Thinking

As they develop their ability to think statistically, sixth-graders will learn about mean, median, and mode, and they will
start describing data distributions. They will also learn about measures and variability and
the effect of outliers.

Overview of Topics

From the four critical areas of focus discussed in the previous section, Common Core also further clarifies the skills sixth-graders
should know by the end of the school year. For example, the fluency requirement at this level
is multi-digit division and multi-digit operations with decimals. The five topics presented here,
taken directly from CCSS itself,
2 include some specifics on what kids will be taught in Grade 6.

Ratios and Proportional Relationships

Understand ratio concepts and use ratio reasoning to solve problems.
The concept of ratios will be taught; for example, every soccer team in the league has 12 players, so the ratio of players
to teams is 12:1. From this concept is introduced the idea that a unit rate of a ratio a:b
is the same as a/b (e.g., if the soccer team has 2 goalkeepers out of its 12 players, 1/6
of the team are goalies). Once ratios are understood, students will solve real-world math
problems including ratios of time and speed, unit pricing, percentages, and measurement.

Gain familiarity with factors and multiples.
Prime numbers are also introduced.

Generate and analyze patterns The idea here is that fourth-graders will recognize
the patterns apparent in the four basic math operations, as well as create patterns based
on a given rule.

The Number System

Apply and extend previous understandings of multiplication and division to divide fractions
by fractions.
This is exactly how it sounds: Students will become proficient in dividing fractions by fractions. Then, they will apply
this concept to word problems.

Compute fluently with multi-digit numbers and find common factors and multiples.
By the end of sixth grade, students complete their fluency of the four basic operations and will be able to add, subtract,
multiply, or divide any multi-digit number, either whole or including decimals. Also, they
will learn to find the greatest common factor (GCF) of two numbers of 100 or less and the
least common multiple (LCM) of two numbers not greater than 12.

Apply and extend previous understandings of numbers to the system of rational numbers.
The concept of negative numbers are emphasized, particularly in real-world quantities (e.g., temperature, budgets, and so
on). Rational numbers and absolute value are introduced, as well as the role of 0 on the
number line. Finally, word problems will include graphing points in all four quadrants of
a coordinate plane.

Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.
Among the subjects taught:

Students will begin to identify and evaluate expressions that replace letters for numbers
(e.g., add x to 8 to get x + 8).

They will identify parts of an expression using the terms they have learned from previous
grades (e.g., quotient, factor, sum, and so on).

They will create and evaluate expressions using whole-number exponents.

They will apply the order of operations to equations, especially those without parentheses.

They will apply the distributive property to reach equivalent equations. For example,
2 (x + 4) is the same as 2x + 8.

They will be able to identify when equations are equivalent. For example, x + x + 2x
is equivalent to 5x - x.

Reason about and solve one-variable equations and inequalities.
With these algebraic concepts down, sixth-graders will begin to solve simple equations and inequalities that include one
variable. They will also use substitution to determine if an equation is true.

Represent and analyze quantitative relationships between dependent and independent variables.
Students will solve problems that use variables to represent two quantities that change in relationship to one another. For
example, how far would a train that goes 50 miles per hour travel in a certain number of
hours? If d represents distance and t equals time, then d = 50t. The equation can change
if the train goes faster or slower.

Geometry

Students will solve for the area of right triangles, special quadrilaterals, and other
polygons by composing the shapes into rectangles or decomposing them into triangles
and other shapes.

The will learn how to find the volume of a right prism.

They will draw polygons in the coordinate plane based on given vertices, and they will
apply this concept to word problems.

They will represent three-dimensional problems using nets of rectangles and triangles
to find surface area.

Statistics and Probability

Develop understanding of statistical variability.
Students will be taught the definition of a statistical question as one that anticipates variability (for example, asking
someone’s age with the knowledge the answer could be within a range of numbers). Also, they
will learn that data collected to answer a statistical question has a center (mean, median,
and mode), a spread (interquartile range, mean absolute deviation, and outliers), and an
overall shape.

Summarize and describe distributions.
Sixth-graders will display data on a number line, including dot plots, histograms, and box plots. Furthermore, they will
summarize data sets in relation to context, including by reporting the number of observations,
describing the nature of the attribute (e.g., how it was measured), by mean and median,
and by identifying patterns in relation to the center.

The Truth About CCSS and Performance

Common Core aims to improve educational performance and standardize what students should learn at every grade in preparation
for a lifetime of application, but it does not set curricula, nor does it direct how teachers should
teach. As with any educational reform, some teachers, schools, and school districts will struggle
with CCSS, some will seamlessly adapt, and some will thrive. As a parent, your responsibility is
to monitor what your sixth-grader is learning, discover what is working or isn’t working for your
child, and to communicate with his or her teacher—and to accept that your children’s math instruction
does differ from what you learned when you were younger, or even what they might have learned last
year. The transition can be a little daunting for parent and student alike, but that’s not a product
of the standard itself. Common Core simply takes a new, more pointed approach to improving the quality
of math instruction in this country.

The Benefits

As previously mentioned, CCSS decreases the number of topics students learn at each grade. However, the remaining topics
are covered so extensively that the chances a child will master the corresponding skills increase.
An analogy to this approach is comparing two restaurants. One restaurant has a varied menu with dozens
of items; the other only serves hamburgers, fries, and milk shakes. The quality of the food at the
first restaurant may vary upon the cooks’ experience, the multitude of ingredients required for so
many offerings, and the efficiency (or lack thereof) of the staff. Because the second restaurant
only serves three items, mastering those three items efficiently should result in an excellent customer
experience. That’s not to say the first restaurant won’t succeed (because many do), but there’s always
a chance that something on the menu won’t live up to the business’s own expectations.

By reducing the number of math topics taught, Common Core helps ensure students are truly ready for what
comes next. Given the attention given to the included concepts, more practical applications and alternate
operations of the math can be explored.

Coinciding with the reduction of topics is an emphasis on vigor—achieving a “deep command” of the math being taught. Students
will be challenged to understand the concepts behind mathematical operations rather than just resorting
to rote memorization and processes to get a right answer. Speed and accuracy are still important;
kids won’t be getting away that easily from flash cards and quizzes that increase fluency. Moreover,
Common Core places even additional emphasis on practical application—after all, the math kids learn
now will be important when they become adults, even if they never have to think about prime numbers
or symmetrical lines in their day-to-day lives.

Finally, CCSS links standards from grade to grade so that the skills learned at one level translate into the tools they need
to learn at the next level. This coherence would seem an obvious educational approach, but often,
there is no link—students are taught a skill in sixth grade that might not be used (and might have
to be re-taught) until eighth. Each new concept in Common Core is an extension of a previous, already
learned concept.

2Grade 4: Introduction, Common Core State Standards Initiative

Math Practices to Help Improve Performance

In addition to the grade-specific standards it sets forth, Common Core also emphasizes eight “Standards of Mathematical Practice”
that teachers at all levels are encouraged to develop in their students.3 These eight practices,
designed to improve student performance, are described here, with added information on how they apply
to sixth-graders.

Make sense of problems and persevere in solving them.
Students explain the problem to themselves and determine ways they can reach a solution. Then, they work at the problem until
it’s solved. This CCSS math practice encourages students to take their time to read and try
understanding the problem, emphasizing that the process is ultimately important even if it
doesn’t result in a correct answer. Sixth-graders, for example, might create a table of numbered
pairs to better understand a ratio. Students this age will also be encouraged to visualize
the problem in order to better reach a solution.

Reason abstractly and quantitatively
Students decontextualize and contextualize problems. By decontextualizing, they break down the problem into anything other
than the standard operation. By contextualizing, they apply math into problems that seemingly
have none. For example, sixth-graders may decontextualize by drawing a number line to
solve an equation involving negative numbers. Kids this age who are contextualizing may
add variables into an equation that doesn’t otherwise have any.

Construct viable arguments and critique the reasoning of others
Students use their acquired math knowledge and previous results to explain or critique their work or the work of others.
Sixth-graders may be asked to work in teams to solve a problem or set of problems, then
present their work to the class.

Model with mathematics
This is just like it sounds: Students use math to solve real-world problems. Sixth-graders can be challenged to take the
math skills they have learned into their own lives. For example, a student may use ratios
and variable equations to determine how many lawns could be mowed in 30 hours if it takes
6 hours to mow 4 lawns.

Use appropriate tools strategically
Another self-explanatory practice: Students learn and determine which tools are best for the math problem at hand. Fourth-graders
might be directed to fgure out the perimeter of their classroom and be given a choice of
a yardstick, a 6-inch ruler, or a tape measurer to achieve that goal. They then decide which
will work best toward a solution.

Attend to precision
Students strive to be exact and meticulous—period. A great thing about ratios and variable equations: Answers can be easily
and efficiently double-checked, and sixth-graders will be expected to review an answer to
ensure they were correct. Here’s another example of the importance of precision: Students
can use the distributive property to express the sum of two whole numbers between 1 and 100
with a common factor as a multiple of a sum of two whole numbers with no common factor—e.g.,
36+8 can also be expressed as 4 (9+2).

Look for and make use of structure
Students will look for patterns and structures within math and apply these discoveries to subsequent problems. For example,
solving for x in an equation might be as simple as adding or subtracting the variable from
each side—a basic algebraic principle that will help sixth-graders with variable expressions.
Another example: The smaller number in negative inequality (e.g., -7
< -3) will always be to the right of the larger number on the number line, which is the reverse of a positive inequality

Look for and express regularity in repeated reasoning
Students come to realizations—“a-ha” moments is a good term for these realizations—about the math operations that they are
performing and use this knowledge in subsequent problems. For example, a fourth-grader
may realize that whenever an odd number is divided by an even number, there will be a
remainder, which is something he or she can look for in future division problems.

Look for and express regularity in repeated reasoning
Students come to realizations—“aha” moments is a good term for these realizations—about the math operations that they are
performing and use this knowledge in subsequent problems. For example, a sixth-grader
might realize that 30 percent of a quantity is the same as 30/100 times the quantity.
He or she can then use that knowledge when solving other problems involving percentages.

How to Help Your Children Succeed Beyond CCSS

Some of parents’ trepidation with Common Core isn’t so much with the guidelines themselves, but with the testing now aligned
with CCSS via local math curricula. Standardized testing was stressful for students and parents
before; with the ongoing Common Core implementation, many families simply don’t know what to
expect.

Fortunately, CCSS does not have to be that stressful, for you or your fourth-grader. Here are some tips to help your children
succeed with Common Core math:

Be informed; be involved

If Common Core concerns you, intrigues you, or confuses you, don’t hesitate to learn as much about it—in your child’s classroom,
at your kids’ school, and on a national level. Talk with teachers, principals, and other parents.
Seek advice on how you can help your kids, and yourself, navigate CCSS math. If you want to take
further action, become involved with PTA or other organizations and committees that deal with
your school’s curriculum. The more you know, the more, ultimately, you can help your child.

Give them some real-world math

A basic tenet of Common Core is to apply math principles to real-world situations. Why not start
now? Give your child math problems when you are out and about— the grocery store, in traffic,
the park, and so on. For example, if you are putting gasoline into your car, before you start
dispensing the fuel, ask your fourth-grader how much money will be required to fill up your 15-gallon
tank. Without a pencil and notebook to compute the answer, he or she might have to fall back
on alternative math processes—processes that Common Core encourages—for a solution.

Take time to learn what they are learning

You might look at a worksheet your child brings home and think, “This isn’t the math I’m used to.”
Because Common Core emphasizes understanding the process of arriving at an answer, your child
may be taught additional ways to fry a mathematical egg, so to speak. Instead of shunning these
approaches, learn them for yourself. Once you comprehend these additional methods, you will be
better able to help your child comprehend them as well.

Encourage them to show their work

This suggestion can be read two ways. First, students will be encouraged to show how they arrived at an answer (and beginning
with fourth-grade math, some answers can be self-checked to see if they are correct), especially
within Common Core. Second, ask your children to show you their homework, particularly the challenging
stuff. Explaining how a problem is solved is a basic CCSS tenet, so if your kids can be confident
in explaining their work to you, they will carry that confidence into the classroom when the
teacher asks for those same explanations.

Seek more help if necessary

If your fourth-grader is struggling with the new math standards, talk with his or her teacher first. You then might want
to seek outside resources to help your child. Several online resources provide math help, including
worksheets and sample tests that conform to Common Core standards. Tutoring might be an option
you consider as well. Innovative iPad-based math programs have emerged that combine the personalized
approach of a tutor with today’s technology. This revolutionary approach also may feature a curriculum
based on Common Core, thus ensuring your child’s learning at home is aligned with what he or
she is learning at school.

Math Practice Worksheets

Ratio and Proportion

Enter the ratio of number of boxes to number of cupcakes as a fraction in its simplest
form: 3 boxes for 15 cupcakes

Identify all the equivalent ratios: For every 4 students who like Math in Victoria Public
School, 8 like Chemistry.

1:2

2:4

8:16

20:10

12 laps in 3 hours = _ laps per hour

Steven saved $54 in 3 days. If he saved the same amount each day, how much money did
he save in a day?

Are these ratios equivalent? 42 books for every 6 notepads 126 books for every 18 notepads

Yes

No

Write unit rate as a fraction in its simplest form Mason traveled 164 miles in 8 hours.
If he traveled the same number of miles each hour, how many miles did he travel per
hour?

Select the car that is the most efficient

Car A

Car B

Car C

Fill in the missing numbers
a = _
b = _
c = _

If the value of x is 10, the value of y is _ .

The Number System

Select the even number smaller than 5 and greater than -18.

-9

-6

12

-5

Identify the coordinates of the point Z.

(2, -7)

(-2, 7)

(-7, -2)

(-2, -7)

If the cross is reflected over the y-axis, give the coordinates of its reflected position.

When the point B is reflected over the y-axis, in which quadrant will the reflected point,
B’, lie?

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Fill in the missing number The distance between (4, -2) and (4, 3) is _ units.

On the map of Jack’s neighborhood, Jack’s home is located at (5, 2). The park is located
3 units right of Jack’s home. What are the coordinates of the park?

Which is the farthest from Myra’s home?

Mall

Park

Hospital

T4 2/5 ÷ 3 7/9

The Highest Common Factor of 24 and 36 is _.

A set of tiles are put as a border of a room. Every third tile in the border is green
and every fifth tile is checked. Which would be the first checked green tile?

Expressions and Equations

Express in an equation form: If 8 is subtracted from a number, the answer is 14.

x - 14 = 8

x + 8 = 14

x + 14 = 8

If a = -14 and b = 6, what is a x b?

Find the value of the expression, when x = 15. (6 ÷ 2)x - 4

Identify all the equivalent expressions

2p + p

6p ÷ 2

p + p + p + p + p

6 × p

If (30+20) is expressed as 10(2+3), where 10 is the greatest common factor of 30 and
20, how can you express (24+28)?

Study the table and define a rule for the output

y = 3x

y = x² + 2

y = 2x + 2

y = x²

Mason bought toffees for $3 and z candies costing $0.25 each. Which expression gives
the total amount spent by Mason?

0.25 × z - 3

0.25 × z × 3

3 × z + 0.25

0.25 × z + 3

Select all the correct options: A shop sells a DVD for $8 and a CD for $4. Select the
correct combinations that Mia would be able to buy if she has $40 dollars in hand.

3 DVDs and 5 CDs

2 DVDs and 6 CDs

4 DVDs and 2 CDs

5 DVDs and 1 CD

Is this statement true or false? When n is an integer less than 0, 10n > 1.

True

False

Select all the values that would make the inequality true: 15 - 4z > 6

2

-1

-3

0

Geometry

Robert painted the walls and ceiling of two of his rooms. Each room measures 15 ft ×
20 ft × 12 ft. How many rectangular faces did he paint?

Does the net match the solid figure of a triangular pyramid?

Yes

No

Which solid figure does the net represent?

Scroll down to see all the options: Which other faces have the same area as B? Select
all the correct options.

A

F

C

E

What is the surface area of the net, in square inches, if each small square measures
1 square inch?

A rectangular pyramid and its net are given. Fill in the missing number. The area of
the net is _ cm².

The area of one of the faces of a closed rectangular prism is 90 cm². The area of another face is 126 cm². If the area of
the base is 35 cm², what is the surface area of the prism in cm²?

Emily is building a tent in the shape of a square pyramid. The area of the base is 25 ft² and the side and height of each
of the triangular faces is 5 ft. How much material, in ft², does Emily need to build
the tent?

Perimeter of a rectangle is 58 cm and one side is 14 cm. Calculate the area in sq. cm.

A cube and its net are given. Fill in the missing number. The area of the net is _ ft².

Statistics and Probability

50 students of grade 5 are asked their ages. In which of these statements can you anticipate
more variability?

What is your age in months?

What is your age in months?

Stella collected the price of DVDs sold at the neighborhood store. The dot plot represents
the data she collected. Describe the shape of data distribution.

The distribution is symmetrical

The data is clustered

The data has an outlier.

The distribution is skewed to the left

Identify the sampling method that the statement conveys:
Andrew asks five members of every age group attending the gym about the number of
hours in a week they exercise.

Biased

Random

Representative

The Basketball team of Brentwood High School played 5 games in a tournament and their
scores are depicted in the Box and Whisker plot. What was their highest score?

The sizes of farms (in acres) in Sonoma County and Ben Hill County are depicted in the
histograms. There are _ more 50-acre farms in Sonoma County than in Ben Hill County.

An ice-cream store recorded the different types of icecreams sold in a day. The table
shows the data. How many Vanilla ice-creams were sold?

Holdings Inc. launched a new website. The number of hits on the first 5 days are listed
here: 35, 95, 120, 115, 110
What is the effect of the outlier on the mean?

The outlier increases the mean

The outlier decreases the mean.

The outlier has no effect on the mean.

Fill in the missing number The time (in seconds) taken by 10 contestants to complete
a puzzle was recorded.

52, 48, 59, 62, 54, 135, 51, 59, 62, 58

The mean increases by _ seconds when outlier is included.

The line graph represents the price of stationery items at a departmental store.

Mean

Median

Mode

Eva is taking French classes. The following are the number of new words learned by her
over a period of 10 days.

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